Methods for the prediction of effective properties of metal foams

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Universidade de Aveiro Departamento de Engenharia Mecânica 2016

José Miguel

Redondo de Aquino

Methods for the prediction of effective properties

of metal foams

Metodologias de previsão de propriedades efetivas de

espu-mas metálicas

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Universidade de Aveiro Departamento de Engenharia Mecânica 2016

José Miguel

Redondo de Aquino

Methods for the prediction of effective properties

of metal foams

Metodologias de previsão de propriedades efetivas de

espu-mas metálicas

Dissertação apresentada à Universidade de Aveiro para cumprimento dos re-quisitos necessários à obtenção do grau de Mestre em Engenharia Mecânica, realizada sob orientação científica de Prof. Doutor João Alexandre Dias de Oliveira, Professor Auxiliar do Departamento de Engenharia Mecânica da Universidade de Aveiro, e de Doutora Isabel Maria Alexandrino Duarte, Investigadora Auxiliar do Departamento de Engenharia Mecânica da Uni-versidade de Aveiro.

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o júri / the jury

presidente / president Prof. Doutor António Gil D’orey de Andrade Campos

Professor Auxiliar da Universidade de Aveiro

Prof. Doutor Nuno Ricardo Maia Peixinho

Professor Auxiliar da Universidade do Minho

Prof. Doutor João Alexandre Dias de Oliveira

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agradecimentos / acknowledgements

Ao Professor Doutor João Alexandre Dias de Oliveira, incansável, sempre disponível, motivador e com uma capacidade de trabalho incrível. É sem dúvida, a vários níveis, um exemplo para mim. À Doutora Isabel Maria Alexandrino Duarte, por me ter introduzido a este tema, e por todo o apoio e motivação que me deu. Foram a melhor orientação que podia pedir um gigante obrigado por todo o esforço e dedicação.

Ao Professor Doutor Matej Vesenjak do departamento de engenharia mecânica da Universidade de Maribor, por todos os conselhos, foram in-questionavelmente um contributo importante para o desenvolvimento do trabalho.

À minha família, em especial aos meus pais Isabel e Luís, por me terem apoiado sempre, pela paciência, pela força e por serem os responsáveis de eu ser quem sou. À pessoa mais importante da minha vida, o meu mano e gémeo João, que me aturou em todas as fases do meu percurso. E à minha tia Ana por todo o apoio, um grande obrigado.

Aos membros do GRIDS, a todos os colegas e funcionários do Departa-mento de Engenharia Mecânica da Universidade de Aveiro pelo apoio e ajuda prestados.

Ao BEST Aveiro onde fiz amizades para a vida. Por todas as experiencias que me proporcionaram. É inexplicável a forma como me integraram, me fizeram evoluir como pessoa e me desafiaram constantemente a superar-me. À Sara pela paciência, força e apoio incondicional. A todos os meus amigos, pessoal da Figueira e de Erasmus, nunca me vou esquecer dos momentos que passámos e espero que venham muitos mais.

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keywords Metal foams; Homogenization; Representative unit-cells; Closed-cell; Open-cell; Elastic behaviour; Effective properties; Kelvin structure; Weaire-Phelan structure; Analytical models

abstract Given their unique properties, there is an increasing interest in using metal foams. In order to expand the usage of these materials, there is a great need to accurately characterize their effective properties. However, there is a great difficulty in predicting the properties of these inhomogeneous mater-ials due to their irregularities and micro defects. The scope of this work is precisely to find analytical models or numerical methods that can describe the behaviour of metallic foams in an elastic regime. To do this, numerical methods and analytical models provided by previous works, were used. To apply the numerical methods, it was necessary to model representative unit-cell geometries. Based on previous works results, the selected geometries were the Kelvin and the Weaire-Phelan structures. With these it was pos-sible to model closed-cell and open-cell representative unit-cells. The open and closed-cell geometries were then subjected to three numerical methods, symmetry boundary conditions with a prescribed force, symmetry boundary conditions with an imposed displacement and asymptotic expansion homo-genization. The two methods that use symmetry boundary conditions were analysed in Femap software and the Asymptotic Expansion Homogeniza-tion, which uses periodic boundary conditions, was analysed with main-FRAN program. It is known that the relative density is the characteristic that has bigger influence on metal foams stiffness. As the analytical models relate the relative Young’s modulus with the relative density, in this work this relation was also obtained for each numerical method. The numerical results were then compared to the analytical models and to experimental results.

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palavras-chave Espumas metálicas; Homogeneização; Células representativas unitárias; Célula fechada; Célula aberta; Comportamento elástico; Propriedades efetivas; Estrutura de Kelvin; Estrutura de Weaire-Phelan; Modelos analíti-cos

resumo Dadas as suas propriedades únicas, existe um interesse crescente em utilizar espumas metálicas. De forma a globalizar a utilização destes materiais, há uma grande necessidade de caracterizar com precisão as suas propriedades efetivas. Contudo, há uma grande dificuldade em prever as propriedades destes materiais não-homogéneos devido às suas irregularidades e micro-defeitos. O âmbito deste trabalho é precisamente encontrar modelos analíti-cos ou métodos numérianalíti-cos que consigam descrever o comportamento das espumas metálicas em regime elástico. Para isso foram usados métodos numéricos e modelos analíticos providenciados por trabalhos precedentes. Para aplicar os métodos numéricos foi necessário, modelar as geometrias das células unitárias representativas. Com base nos resultados de trabal-hos já existentes, as geometrias selecionadas foram as estruturas de Kelvin e de Weaire-Phelan. Com estas geometrias definidas, foi possível mod-elar células representativas unitárias de célula aberta e de célula fechada. Após definidas, as geometrias de célula aberta e célula fechada foram sub-metidas a três métodos numéricos, condições de fronteira de simetria com uma força prescrita, condições de fronteira de simetria com deslocamento imposto e homogeneização por expansão assimptótica. Os dois métodos que usam condições de fronteira simétricas foram analisados no programa Femap, o procedimento de homogeneização por expansão assimptótica, que usa condições de fronteira periódicas, foi analisado através do programa mainFRAN. Sabe-se que a densidade relativa é a característica que tem maior influência sobre a rigidez das espumas metálicas. Como os modelos analíticos relacionam o módulo de Young relativo com a densidade relativa, neste trabalho esta relação também foi obtida para cada método numérico. Os resultados numéricos foram então comparados com modelos analíticos e com resultados experimentais.

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List of Tables

2.1 List of parameters for describing the structure of metallic foams. . . 7

5.1 Relative density distribution for each type of RUC . . . 37

5.2 Element types used. . . 38

5.3 Characteristics of the components . . . 45

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List of Figures

1.1 The different manufacturing methods. . . 2

1.2 Properties and applications of the closed and open-cell foams . . . 3

2.1 Closed-cell foams. . . 5

2.2 Open-cell foams. . . 6

2.3 Design Variables. . . 8

2.4 Representation of a stress-strain curve of a metal foam. . . 8

2.5 Representation of the deformation mechanisms that contribute to linear-elastic response of foams. . . 9

2.6 Comparison between two foams with different pore size, a) has bigger pore size and smaller density than b) . . . 11

3.1 Overview of the foam representative models. . . 13

3.2 Representation of a kelvin cell (left) and the two different polyhedra that constitute the Weaire and Phelan structure (middle and right, respectively). 15 3.3 Voronoi Tessellation representation of seed points. . . 15

3.4 A macroscopic body of size LM with a mesoscale window of size L, in which a microstructure of size d is shown. . . 16

3.5 An representation of a composite and equal size alternatives RVE (windows). 19 3.6 Representation of an original RUC and the respective periodic deformation. 24 3.7 Representation of the application of symmetry boundary conditions to a unit-cell. . . 24

5.1 Work-flow for the numerical methodology. . . 33

5.2 The representation of the Kelvin geometry . . . 34

5.3 The three faces that compose the Weaire-Phelan structure. . . 35

5.4 The figure compiles all RUCs type used in this work. . . 35

5.5 Foam representation (left side) and the block parent material (right side) 36 5.6 A representation of the overlapping material when using beam finite ele-ments . . . 36

5.7 Five refinement levels, (a) coarse, (b) medium coarse, (c) medium, d) medium fine and e) fine. . . 38

5.8 Constraints for the different cases (a), (b) and (c) . . . 39

5.9 Three types of loads applied, (a) force applied on the upper surface, (b) force applied on a non-deformable plate (c) enforced displacement of the upper face. . . 40

5.10 The convergence curves corresponding to each case and for each level of mesh refinement (on the right side) and the average curve (on the left side). 40 5.11 The average convergence curves for each type of Kelvin RUCs. . . 41

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5.14 The two considered composite materials, (a) Parallel association and (b) serial association . . . 44 5.15 Comparison of the Young’s modulus obtained from the numerical

proced-ure with the rule of mixtproced-ures. . . 46 5.16 Open Weaire-Phelan RUC, on top 1) without second phase, on bottom

2) with a defined second phase. . . 46 5.17 Loading vectors a), b) and c) the normal modes and d), e) and f) the

shear modes. . . 47 5.18 Predicted relative Young’s E/Es modulus by the AEH method, with and

without a defined second phase, in comparison to the other methods. . . . 47 5.19 The influence of the number of cells in a periodic microstructure (for a

Kelvin RUC) . . . 48 5.20 The example of periodic displacement of a RUC submitted to the AEH

(a) representation of the PMS ,(b) for a normal mode, (c) and for a shear mode . . . 48 5.21 The periodic microstructures . . . 49 6.1 Closed-cell periodic microstructures deformation when subjected to SPF

method. (a) Kelvin solid, (b) Kelvin surface, (c) Weaire-Phelan solid and (d) Weaire-Phelan surface. . . 52 6.2 Predicted Relative Young’s modulus for an prescribed uni-axial force and

symmetry boundary conditions. . . 53 6.3 Relative deviation between the analytical models and this method results. 53 6.4 Closed-cell periodic microstructures deformation when subjected to SID

method. (a) Kelvin solid, (b) Kelvin surface, (c) Weaire-Phelan solid and (d) Weaire-Phelan surface. . . 54 6.5 Predicted Relative Young’s modulus for an imposed uni-axial

displace-ment and symmetry boundary conditions. . . 55 6.6 Relative deviation between the SPF and SID methods . . . 55 6.7 Predicted Relative Young’s modulus for AEH . . . 56 6.8 Characteristic displacements for a Kelvin unit-cell, for the normal modes

a) χxx, b) χyy and c) χzz and the shear modes d) χxy, e) χyz and f) χxz. . 57

6.9 Characteristics displacements for a Weaire-Phelan RUC, in the normal directions a) χxx, b) χyy and c) χzz and the shear modes d) χxy, e) χyz

and f) χxz. . . 57

6.10 Predicted Relative Young’s modulus for AEH using linear tetrahedrons elements . . . 58 6.11 Results for the changes in the different parameters. . . 59 6.12 Open-cell microstructures deformation when subjected to SPF, (a) Kelvin

Solid OP, (b) Kelvin Wire, (c) Phelan Solid OP and (d) Weaire-Phelan Wire. . . 60 6.13 Predicted Relative Young’s modulus for an prescribed uni-axial force and

symmetry boundary conditions. . . 60 6.14 Relative deviation between the closed formulation and this method results. 61

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6.15 Open-cell microstructures reaction when submitted to SID method, (a) Kelvin Solid OP, (b) Kelvin Wire, (c) Weaire-Phelan Solid OP and (d) Weaire-Phelan Wire. . . 62 6.16 Predicted Relative Young’s modulus for an imposed uni-axial

displace-ment and symmetry boundary conditions . . . 62 6.17 Relative deviation between the SPF and SID methods. . . 63 6.18 Predicted Relative Young’s modulus for HEA for linear tetrahedron

ele-ments (lte) and quadratic tetrahedron eleele-ments (qte). . . 63 6.19 Characteristics displacements for a kelvin open-cell RUC, in the normal

modes a) χxx, b) χyy and c) χzz and the shear modes d) χxy, e) χyz and

f) χxz. . . 64

6.20 Characteristics displacements for a Weaire-Phelan open-cell RUC, in the normal modes a) χxx, b) χyy and c) χzz and the shear modes d) χxy, e)

χyz and f) χxz. . . 64

6.21 Comparison of the relative Young’s modulus given by the closed-cell for-mulations to the experimental results. . . 66 6.22 Comparison of the relative Young’s modulus given by the open-cell

for-mulation to the experimental results. . . 66 6.23 Comparison of the relative Young’s modulus given by the SPF and SID

methods to the experimental results. . . 67 6.24 Predicted Poisson’s ratio for open-cell microstructures and comparison to

two closed models. . . 68 6.25 Difference between the Young’s modulus using linear tetrahedral elements

and quadratic tetrahedral elements, for a Kelvin closed-cell RUC and a Weaire-Phelan open-cell RUC . . . 69 6.26 Difference of the anisotropy ratio relation with increasing the density for

the different models. . . 69 6.27 Anisotropy maps of the different cells, the maps on top correspond to the

lowest density cell and on the bottom to the highest density cell. . . 70 6.28 Orientation dependence of the Young’s modulus for the different RUCs. . 71

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Chapter 1

Introduction

Solid metallic foams are a class of materials of increasing interest, that combine a very exciting set of properties that make them interesting for applications in a wide vari-ety of sectors, such as transport, defence, architecture [1]. B. Sosnik [2] performed the first attempt to create a metal foam in 1943, when he added mercury to molten alu-minium in order to obtain pores. In the late 50’s, J. C. Elliot [3] replaced mercury with foaming agents generating gas by thermal decomposition, avoiding the toxicity of the mercury. B.C.Allen [4] invented and patented the Powder Compact Foaming (PCF) method. Aluminium foams was prepared by heating an extruded precursor containing powders of aluminium and blowing agent (i.e. TiH2, ZrH2 and CaCO3) at temperatures

above the melting point. However, the resulting foams had bad quality. The liquid foam was kept during to much time at temperatures above the melting point, leading to a collapse of the formed foam. Allen reports the need of progressive heating of the precursor above the melting point and the fast cooling of the foam after expansion. Nev-ertheless, foams made from extruded precursor was more stable than the ones produced by Elliot. In order to increase the stability of aluminium foams several inventions were published in the late 60’s early 70’s based on the conception of increasing melt viscosity or thickening. At the end of the 80’s, patents and articles appear describing new methods for production, improving significantly the quality of the foams. In 1987, the Japanese Shinko Wire Company Ltd. registered the first aluminum foam trademark or process called Alporas. The foams are prepared by adding a blowing agent (TiH2) into a molten

metal containing a viscosity-increasing agent (i.e. Ca). In 1990, the Canadian Alcan [5] and the Norwegian Norsk Hydro [6] companies patented the direct foaming processes. The foams are prepared by directly injecting a gas (i.e. argon and air) into the molten melt containing ceramic particles (a viscosity-increasing agent). In 1992, Baumeister et al. [7] based his work on Allen’s research, improving the Powder Compact Foaming (PCF) method. Nowadays, these are the methods used to fabricate the closed-cell foam parts at a commercial level. The replication technique, used to fabricate polymer and ceramic foams, has been adapted into the manufacturing of open-cell metal foams firstly in 1966 in order to use them as porous battery electrodes [8]. Nowadays, the open-cell foam parts are usually prepared by burning of a polymer foam that gives place to a ceramic template that is replicated by the metal, originating the metal foam [9–11]. Nevertheless, there are several ways to fabricate metal foams which are usually divided according to the initial state of the metal (powder, molten metal and ionised metal), as shown in Figure 1.1. The properties of the metal foams depend on the manufacturing

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Figure 1.1: The different manufacturing methods [10].

process used. For example, the powder compact foaming method can produce closed-cell foams with size pore range of 2-5 mm and a density within the range 0.2-1 g/cm3, while the replicate processes achieving open-cell foams, achieving porosities in the range of 80% to 97%. Metal foams are divided into two main groups according to their cellular structures: closed-cell foams as the ones where cells (isolated pores) are separated from neighbouring ones by cell walls, and open-cell foams where the cells are interconnected by cell edges or ligaments [12]. The properties, specially the mechanical properties de-pend on: the characteristics of the base material (from which the foam is made), relative density (ratio of the foam density to the density of base material) and morphological parameters (i.e. pore size, type of foam). Both foams are light-weight, non-inflammable and recyclable materials, but the additional properties are quite different (Figure 1.2). To the industrial application, it is important to understand the processing-structure-properties relationships [10, 13, 14]. In these applications, elastic processing-structure-properties and critical failure loads are particularly important. The mechanical properties are in large part a result of the complex cellular structure of foams. The relationships between the proper-ties and the size of the pore and the foam density have been extensively studied using experimental studies. In order to spread the usage of these materials on engineering applications, it is required a detailed understanding of its mechanical properties and behaviour. An extensive number of uni-axial loading, bending, fatigue experiments were already performed to understand the plastic deformation. However, these experimental studies have limitation because foams with regular structures and without defects are hardly available. In general, real foams exhibit a high variability in cell sizes and shapes, and wall thickness that influences their elastic properties. Most publications are focusing on the elastic properties as well as simulate also the elastic plastic material behaviour. Several researchers investigate the influence of geometrical imperfections on the effective properties. To solve it, numerical and analytical studies have been performed as al-ternative methods to predict the macroscopic properties of heterogeneous materials, as foams based on the properties of constituent materials and their morphology. To study the properties-microstructures relationship of foams and failure mechanism at the cell level, foam micromodels have been developed over the last decade, which basically fall

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1.Introduction 3

Figure 1.2: Properties and applications of the closed and open-cell foams

into three categories: space-filling polyhedron, tessellation-based and image-based mod-els. Relevant literature shows three main modelling approaches for three-dimensional Finite Element (FE) models of cellular materials. One considers unit-cell models, which include a limited number of cells and are based on a one initial geometry in which the geometry is repeated periodically. Some authors focus on the irregularities between cells and the disorder in the microstructure. They generate models, for example by a Voronoi tesselation, with different cell size distributions. These models are based on the second method for a numerical setup. Other authors pursue the third kind of models, using volume image data for the implementation of the cellular microstructure to generate representative volume elements which reproduce the real microstructure almost exactly. The combination of microstructure models with finite element simulations is a well-established approach for studying the relationship between the microstructure geometry and the macroscopic properties of a material.

1.1 Objectives

The increasing interest in metal foams leads to a necessity of properties characterization. This work focuses on studying methods for effective predictions of metal foams, and has

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the following main objectives:

• To define procedures to characterize the linear elastic behaviour of metal foams; • To study the structural parameters that affect the mechanical behaviour of metal

foams;

• To model representative geometries of foams and apply different homogenization methods using numerical analysis software;

• To compare the predicted results from the numerical methods with models from previous works, and evaluate how the predicted behaviour approximate experi-mental results.

1.2 Document Outline

This work is organized into 7 chapters and the references. The first Chapters, 1 to 4 are related to the theoretical part, while Chapters 5 and 6 with the numerical procedure and results discussion:

• Chapter 1 provides an introduction to the research work presented in this disser-tation, describing its outline;

• Chapter 2 gives an overview of the main properties of the metal foams, describing their typical compressive behaviour and what factors affect it;

• Chapter 3 works as the bridge to the numerical modelling. First by exposing the different types of foam models used in previous works, secondly by introducing the numerical methods for the material characterization, and the state-of-the-art for the application of these methods on cellular materials;

• Chapter 4 introduces the early studies for predicting foams behaviour, and as-sembles some of the models from previous works;

• Chapter 5 presents the numerical procedure is presented, from the modelling of the foams to the validation of the methods;

• Chapter 6 shows the predictions from the different methods are compared between them, the models introduced in the Chapter 4 and the experimental results. There is also a study on the orientation dependence of the Young’s modulus for the representative unit-cells;

• Chapter 7 gives an overall appreciation of the work and its objectives, compiles the major outcomes and also some thoughts on prospects for future work

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Chapter 2

Foam Properties

2.1 Cellular Solids

The production method of a foam defines its microstructure and usually involves a con-tinuous liquid phase that solidifies. These kind of foams are known as solid foams. Their topological and geometrical properties are identical to liquid foams. When evaluating the bubble growing process, the liquid bubbles solidify and give place to the solid pores. In the liquid state, liquid metal is drained from the cell walls and distributed into the edges minimizing the surface energy and creating a polyhedral geometries. The resultant foam can be either a closed-cell (dry or wet foam) or an open-cell foam [12].

2.1.1 Closed-Cell Foams

Closed-cell foams consist in a structure where the cells are divided by walls between the cell edges (Figure 2.1). Even though some cell walls may have defects such as cracks and micropores, the whole structure is efficiently impermeable to fluids [9,15,16].

Figure 2.1: Example of closed-cell foams [9].

2.1.2 Open-Cell Foams

Open-cell foams are composed by a connected network of struts rigidly bound at their joints, usually with interconnected voids. The manufacturing method, uses a mould where the liquid metal is poured, after the metal solidify the mould is removed, leaving only the borders/struts. According to Banhart [9], the term metal sponges is more

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appropriated to classify these cellular metals. However this designation is not commonly used since real sponges open-cell foams are permeable to fluids. Figure 2.2 shows an example of open-cell foams.

Figure 2.2: Example of open-cell foams [17].

2.2 Foam Structure

Like composites materials, foams are inhomogeneous materials, which means that ma-terial properties are determined by the properties of the constituents, by their geometry and topology. The geometric structure or architecture is defined by the shape and ar-rangement of the cells that are embedded in the metal.

When zooming on the cell material, it is visible another complex substructure com-posed of micropores and cracks. This shows that metal foams are hierarchically struc-tured. Therefore, when describing cellular metals it is important to define what different structural features have impact on the material properties. Table 2.1 shows a list of para-meters for describing the geometrical structure and microstructure [11].

2.3 Relative Density

One of the most important structural characteristics of cellular materials has to do with the density distribution. It is not usually determined in quantitative structural analysis because it is influenced by many structural features [11]. The scaling model of Gibson and Ashby [16] suggests that the mechanical properties of an ideally homogeneous foam are scaled by ¯ρn_{, where n is a constant that depends on the mechanisms governing the}

deformation of the cell walls and ¯ρ is the relative density. The relative density can be calculated by dividing the effective density of the cellular material ρ by the density of the solid massive material ρs,

¯

ρ = ρ

ρs

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2.Foam Properties 7

Table 2.1: List of parameters for describing the structure of metallic foams [11].

The volume fraction of the pores is inversely proportional to the relative density. Therefore, the higher the volume fraction of pores in a material, the lower is the relative density, namely ¯ ρ = ρ ρs = Ms Vt Ms Vs = Vs Vt = 1 − p. (2.2)

Where Vs and Ms, are the volume and mass of the solid part respectively, Vt stands for

the total volume and p stands for the material pore fraction.

2.4 Mechanical Properties/Compression Behaviour

This section aims to offer an overview of the metal foams behaviour under a compression test. Compression behaviour has been the subject of various studies whereas studies of the behaviour under tensile and shear loading are limited. Currently, these compression tests are carried out by a standard procedure given by the ISO 13314 [18]. These materials have extensive number properties that affect their overall behaviour (Figure 2.3). However, the scope of the present work will focus on the compression behaviour.

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Figure 2.3: Design Variables [19].

2.4.1 Deformation and Failure Mechanisms

In compression, metal foams show a unique stress-strain response (Figure 2.4). A more profound analysis to this behaviour, leads to understand some of the characteristics and applications of these materials as for example, energy absorbing.

Figure 2.4: Representation of a stress-strain curve of a metal foam.

There is a difference between open-cells and closed-cell foams in the correspondent deformation mechanisms (Figure 2.5). In the open-cell foams, the main deformation mechanism is bending of the cell edges and extension and compression of the edges. For higher relative densities, ρ/ρs > 0.1, as for closed-cell foams, they combine the

deformation mechanisms pointed for the open-cell foams and membrane stresses in the cell walls.

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2.Foam Properties 9

Figure 2.5: Representation of the deformation mechanisms that contribute to linear-elastic response of foams [16].

Linear Elasticity

The Young’s modulus and the Poisson’s ratio are directly related with the first stage or elastic part of the stress-strain curves corresponding respectively to the line slope and to the relation between the transverse strain and the compression axis strain. Deformation mechanisms are dependent on the type of cell, open or closed, the material which the foam is made of and the density of the foam. Usually metal foams show a considerable ductile behaviour. A compression loading, at small strains, applied to a ductile foam, yields to extension/compression of the cell walls and edges [11,20]. The Young’s modulus, E, the shear modulus, G, and the Poisson’s ratio ν scale with density as [21]

E ≈ α2Es _ρ ρs n G ≈ 3 8α2Gs _ρ ρs n ν ≈0.33, (2.3)

where n has a value between 1.8 and 2.2 and α2 between 0.1 and 4 depending on the

structure of the metal foam, s stands for the solid massive material. Elastic Collapse and Plateau Region

Increasing the compression load on the foam causes to buckling of the cell edges and walls in the weaker regions of the foam leading to early plastic deformation or fracture. If the stresses in the edges and walls exceed the yield stress σy of the solid, the onset

of plasticization is reached and the deformation is no longer reversible. Thus the linear elastic part of the stress-strain curve of the foam is, in general difficult to develop. The plastic collapse begins when a deformation band, perpendicular to the loading axis appear, developing the plateau region. Depending on the cellular structure and the properties of the solid this region, can be approximately flat or wavy. Usually, open-cell foams have a long, well defined plateau stress, σpl, the cell edges are yielding in bending.

On the other hand, because the cell faces have membrane stresses, closed-cells show a

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more complicated behaviour and that can lead to the stress to rise with increasing strain. The plateau stress,σpl, and the densification strain, εd, scale with density as:

σpl≈(0.25 to 0.35)σy,s _ρ ρs m εd≈ 1 − α1 ρ ρs . (2.4)

Currently, for metal foams the value m is between 1.5 and 2.0 and α1 lies between 1.4

and 2.0 [21].

The plateau stress performs an essential role to characterize the energy absorbing behaviour, the absorbed energy for unit of volume corresponds to the area defined by the stress-strain curve. It can be expressed as,

˜ U =

Z ε2

ε1

σ(ε) dε, (2.5)

where [ε1, ε2] is the strain interval. Considering an ideal absorber, the material that

maximizes the equation (2.5) which is an ideal plastic material where ˜U = σ(ε2− ε1),

the efficiency of energy absorption is calculated by [11], η=

Rε2

ε1 σ(ε) dε σ(ε2− ε1)

. (2.6)

Plastic Collapse and Densification

The crushing of cells during the plateau region leads to a densification phenomenon. When the cellular material is compressed, the cell walls and edges starts merging leading to an increased stiffness which can be compared to the base solid material. This leads to the final stage of compression, the stress value increases greatly for small strain variations. In the limit situation the stiffness of the compacted material is approximated to the solid massive material. This way, it is possible to obtain the Young’s modulus of the base material by studying the slope of this last segment [10].

2.4.2 Factors Affecting the Mechanical Properties

Effects of Cell Size and Shape

The cell size and shape are among the properties that describe the pore geometry (Table 2.1). Even if the cell representation is in some works regular and polyhedral, in real foams pores are arranged randomly and there are various sizes, shapes and they can also take preferred orientations. Therefore, an approximation is often taken in order to try to describe some of the features of the pores. They can be described by their average length, volume, area, etc.. The same happens for the shape, by studying the average features of the cell shape,i.e. number of edges per face, difference between height and width, etc. [11,22]. It is proven that low density cells have larger pore size (Figure 2.6). It has been shown that the size of the pores, combined with a certain shape can allow deformation bands to start [11]

Effect of Density

Density can be considered as one of the most significant structural properties. A clear relation between relative density, stiffness modulus and compression plateau stress was

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2.Foam Properties 11

Figure 2.6: Comparison between two foams with different pore size, a) has bigger pore size and smaller density than b) [20].

derived from various scaling models, the Gibson and Ashby model [16] being considered the most general. Based on this scaling laws, the relative density has strong influence on the material properties.

Effect of Strain Rate

The studies based on the effect of the strain rate during the compression test are not consensual. Some of the studies prove that metal foams are sensitive to strain rate [20,23–25]. According to those, at higher strain rates (dynamic loading) the compressive strength is increased, others related density with the strain rate showing that foams with higher relative density are more sensitive to strain rate, while foams with low relative density have insignificant strain rate sensitivity. The main mechanisms associated to this are the micro-inertial effects, strain-rate sensitivity and the entrapped gases in closed-cells [20]. However other studies report that the metal foams are insensitive to strain rate. The explanation may be in the material used as the foams are prepared by different manufactures [26–28].

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Chapter 3

Modelling Cellular Materials

3.1 Foam Models

A foam is composed by stacked cells. The morphology of these cells appear, in the literature, in many kinds of shapes and geometries (Figure 3.1). In this section, various types of models used in previous works are discussed, as well as a comparative ana-lysis between morphological studies and representative geometries. The simplest models are the two-dimensional models. 2-D models can be polygons, i.e., triangles, squares, hexagons, etc., they can be stacked in more than one way, isotropically or anisotropic-ally, to fill a 2-D plane [29]. Three dimensional representative models can be spherical or polyhedron models, these can be packed in many ways,i.e., simple cubic, body-centred cubic, randomly etc., and prismatic models despite the uncommon usage. They have the same geometries as the 2D polygons but they are prismatic in a third direction.

Figure 3.1: Overview of the foam representative models.

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To proceed to further models, there is a need to introduce some information based on real foams. Foams result from the nucleation and growth of bubbles, when the foam relaxes, tends to a local minimum surface free-energy density (Em), minimizing

the surface tension (σ) and increasing the surface area (As) per unit volume (V). This

relation can be described as [30],

Em= σ

As

V . (3.1)

At this state of minimum energy the foam is at equilibrium. According to Plateau’s Law (1873) a foam structure is in equilibrium when:

• Three plates always meet at angles of 120◦ _{to form a plateau border;}

• Four plateau borders must join at the tetrahedral angle of cos−1_{(−1/3) ≈ 109.47}◦_;

• The transition from a plateau border to an adjacent plate is smooth.

Another approach to cellular materials was the studies of Matzke [22, 30, 31], who ex-perimentally quantified the morphological characteristics of the foams. He observed 600 liquid bubbles using a microscope and found that the number of cell faces per cell ranges from 11 to 17 with an average of 13.7. This results indicated that, the average cell has 14 faces and the average face has 5 edges, which means in general that the faces are pentagonal.

Based on Plateau’s laws and Matzke studies, it is possible to evaluate the repres-entative solutions to properly characterize their behaviour. A possible solution is the pentagonal dodecahedron: it consists of a 12 faces polyhedron where each face has 5 edges. Even answering to some of the requirements, it lacks the capability to fill space. The rhombic dodecahedron is another possible solution: it has 12 faces, 24 edges and two types of vertices. It fills space with efficiency but it does not respect the Plateaus’s rules, and so it’s not stable, [29, 32, 33]. The tetrakaidecahedron is seen as the best solution: it can fill space and it approximately obeys to Plateau’s laws. Based on these geomet-ries, Lord Kelvin introduced the Kelvin cell: it consists of a tetrakaidecahedron with 14 faces (where six are squares and eight are hexagons) and with slightly curved faces, it is the single unit-cell that minimizes the surface area and maximizes the filled space when stacked, in contrast to Matzke there are none pentagonal faces [34]. Another solution was found by Weaire and Phelan: this periodic structure has smaller surface area than the Kelvin cell (Figure 3.2). Weaire and Phelan found these geometries using a computer software for the minimization of surface area, they found a periodic structure consisting of two different polyhedra (one consists of a 14-side polyhedron, the other is a 12-side polyhedron) of equal volume. Between the two polyhedra there are only three different faces. Briefly there are two unit-cell models that must be taken into account in order to comply with Plateau’s laws, Kelvin and Weaire-Phelan [15,35]. Given their morphology, Plateau’s laws do not apply to open-cell foams [30]. However, the considered geometries can be also used as representative geometries for open-cell foams, where all the material is concentrated at the edges.

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3.Modelling Cellular Materials 15

Figure 3.2: Representation of a kelvin cell (left) and the two different polyhedra that constitute the Weaire and Phelan structure (middle and right, respectively) [12].

There are also the Voronoi diagrams/tessellation. The resulting foam is a realistic representation of foams that result from nucleation and growth of bubbles. Voronoi tessellation distributes seed points randomly in space (Figure 3.3). Around each seed point spherical bubbles grow uniformly. When a contact point between two bubbles is generated, the growth is halted, but continues in the other directions [36–38]. With seed points arranged by random sequential adsorption (RSA) algorithm and random close packing (RCP) algorithm [32, 39], Voronoi tessellations can be constructed with microstructural topology close to Matzke’s observation. However, Voronoi tessellations cannot be produced with cells of size following a prescribed distribution. Laguerre tes-sellation [40–42], a type of weighted Voronoi testes-sellation, is capable of accomplishing so. In Laguerre tessellations, each seed point has a weight, which plays a role in de-termining the size of the cell that encloses the seed point. Provided that the centres of a set of random close packed spheres are taken as the seed points of a Laguerre tes-sellation and the radii of these spheres are chosen as the weights, then the constructed Laguerre tessellation will have a cell size distribution close to the diameter distribution of these spheres. In addition, Laguerre tessellations constructed in this manner have microstructures that agree well with Matzke’s observations. With the development of X-ray Computer Tomography (CT) techniques, foam finite element models based on the reconstruction of real foams using CT techniques have also been reported [23,43–45].

Figure 3.3: Voronoi Tessellation representation of seed points [46].

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3.2 Micromechanics of Materials

3.2.1 Length Scale

Most of the micromechanical models are based on the assumption that the length scales of a material differ substantially. The lowest length being the microscale (d), the largest one the macroscale (LM) and the intermediate ones the mesoscale (L) (Figure 3.4). The

microscale is characterized by having lengths similar to the sizes of the inhomogeneities. The mesoscale is characterized by lengths large enough to be statistically homogeneous. The macroscale can be described as a sample of the body material of length LM[47]. The

fields describing the behaviour of an inhomogeneous material, i.e. mechanical stresses, strains and displacement, are split into contributions corresponding to each length scales, which as referred to as micro- meso- and macrofields, respectively [48]. When considering length scales there are two situations: In the first one, the inhomogeneous material has a low level of disorder/mismatch in the properties, being regular/nearly-periodic the geometrical and material data can be obtained based on the unit-cell. In this case d < L << LM. In the second one, the material possesses a large number of random

inhomogeneities, and the relation L/d increases, when this relations is finite it creates a necessity of defining a mesofield. In this case d << L << LM [49].

Figure 3.4: A macroscopic body of size LM with a mesoscale window of size L, in which

a microstructure of size d is shown [49].

This is understood to imply, on the one hand, the fluctuating contributions to the fields at a smaller length scale (“fast variables”) influence the behaviour at the larger length scale only via their volume averages (these fluctuations can be described as resid-ual stresses and strains, produced without external forces). On the other hand, gradients of the fields as well as compositional gradients at the larger length scale (“slow variables") are not significant at lower length scales, where these fields appear to be locally constant and can be described in terms of uniform applied fields [48, 50]. Formally this splitting of the stress and strain fields into fast and slow contributions can be written as

ε(x) = hεi + ε0(x) and σ(x) = hσi + σ0(x), (3.2) where x is a position vector, hεi and hσi are the macroscopic slow fields, while ε0 _and

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3.Modelling Cellular Materials 17

3.2.2 Representative Volume Element

The exact definition for Representative Volume Element (RVE) is far from being uni-versal. In this subsection the aim is to clarify the difference between RVE and Rep-resentative Unit-Cell (RUC) and to establish a nomenclature for the present work. As has been said according to inhomogeneous materials microgeometries, it is possible to split them into two distinct categories. The first being the representations that involve the concept of statistical homogeneity and the second one the unit cell periodicity based geometries. The RVE approach is directly connected to the statistically homogeneous microstructure1_{. According to the classical definition proposed by Hill [51]:“a sample}

that is structurally entirely typical of the whole mixture on average, and contains a sufficient number of inclusions for the apparent overall moduli to be effectively inde-pendent of the surface values of traction and displacement, so long as these values are macroscopically uniform". In other words the RVE is usually regarded as a volume Ωsof

heterogeneous material that is sufficiently large to be statistically representative of the inhomogeneous material, independently of which is the physical property in study [50], while the RUC is intrinsically linked with the periodic microstructure representing the smallest element. The microstructure, in a periodic case can be described through a representative unit-cell. The confusion between the concepts of RUC and RVE is related to the periodic unit-cell arrangements, when the microstructures are composed by peri-odic arrays represented by a single RUC [52]. For the present work, these will be called periodic microstructures.

3.2.3 Homogenization and Localization

Homogenization is the process that describes the effective behaviour of an inhomogeneous material at the larger length scale using information from a lower length scale. By simulating the overall material response under simple loading conditions, it is possible to obtain the homogenized properties of the inhomogeneous material. This relations between smaller and higher length scales lead to a characterization of an equivalent homogeneous material [48]. Localization can be described as the inverse process of the homogenization. Based on results obtained at a larger length scale, it is possible to describe with precision the local fields behaviour of the material at a smaller length scale. This way is possible to update the behaviour at the lower length scale in function of the local current state in every iteration [53,54].

Being x the position vector, for any the volume element Ωs of an inhomogeneous

material, homogenization relations takes form of volume averages of some variable f(x) [48]: hf i= 1 Ωs Z Ωs f(x)dΩ. (3.3)

Accordingly, the homogenization relation or volume averages for stress and strain tensors can be defined similarly,

hσi= 1 Ωs Z Ωs σ(x)dΩ. (3.4) 1

While in the Chapter 2, the microstructure refers to the solid material, here the microstructure is related to the inhomogeneities geometry and arrangement.

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hεi= 1 Ωs

Z

Ωs

ε(x)dΩ. (3.5)

An important related concept, introduced by Hill [51], is the stress B(x) and strain A(x) concentration tensors. These are essentially the ratios between the average mi-croscopic fields and the corresponding mami-croscopic responses (localization relations). Formally, they can be written as,

σ0(x) = B(x)hσi, (3.6)

ε0(x) = A(x)hεi, (3.7)

The homogenized strain ans stress fields can be linked by effective elastic tensors E∗

and C∗_{, according with Hook’s law,}

hσi= E∗hεi and hεi = C∗hσi (3.8) Using the above relations and the Equations (3.3) and (3.4) these effective tensors can be obtained from the local tensors, A(x) and B(x), as volume averages,

E∗ = 1 Ωs Z Ωs EAdΩ (3.9) C∗ = 1 Ωs Z Ωs CB dΩ (3.10)

For large integration volumes, the volume averages of fluctuations for the Equations 3.1, 3.3 and 3.4 vanish, 1 Ωs Z Ωs σ0(x) dΩ = 1 Ωs Z Ωs ε0(x) dΩ = 0. (3.11)

With the previous condition, where the mean-fluctuations (σ0 _{and ε}0_{) are zero, the}

volume average of the energy density over Ωs,

hU i= 1 2Ωs Z Ωs σ(x)Tε(x) dΩ = 1 2hσTεi= 1 2hσiThεi (3.12) This condition is called the Hill condition or Hill-Mandel Macrohomogenety condi-tion, it means that the average of the scalar product of the stress σ and strain ε tensors (micro level) equals the product of their averages (macro level). [47–49,55,56]

hσTεi= hσiThεi. (3.13)

3.3 Micromechanical Methods for Material

Characteriza-tion

In continuum micromechanics two main principal strategies are suitable to perform ho-mogenization and localization [48]:

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3.Modelling Cellular Materials 19 • Mean Field Approaches and Variational Bounding Methods: These are simple methods that usually describe the behaviour of a material based only on the phase average of each different constituent. The phase geometry is obtained via statist-ical description. The Variational methods are used to obtain the lower and upper bounds of the physical properties of inhomogeneous materials. These approaches are computationally inexpensive when compared with Discrete Microfields Ap-proaches and they are also referred to as "noninteracting approximations" because they do not account for particle interactions. The informations given by these methods are useful, but not accurate, hereupon they are of limited use for predict-ing the behaviour of cellular metals.

• Discrete Microfield Approaches: Use analytical or numerical methods to predict the actual behaviour of the material by simplifying the microgeometries. The three main representatives of these models are: Periodic Microfield Approaches, Embed-ded Approaches and Windowing Approaches. As long as the real microgeometry is simplified and idealized it is possible to use analytical methods for describing the micromechanical behaviour. However these methods have limitations. Con-sequently, and for a more robust analysis, numerical methods are preferable.

3.3.1 Windowing Approaches

In the present work, the numerical model is based on periodic microfield approaches, which means that the representative geometries in use are periodic. Usually unit-cell models can exhibit a considerable degree of anisotropy [15], while random multi-cell micromechanical models (Figure 3.5), also known as “super-cell models", tend towards isotropic elastic behaviour. The aim of the windowing methods is precisely to estimate the bounds for the macroscopic properties of inhomogeneous materials, with non-periodic volume elements. Because windowing methods describe the behaviour of a inhomogen-eous sample rather then of inhomogeninhomogen-eous material, instead of “effective property” it is used the term “apparent property”. This concept was introduced by Huet [55]. These

Figure 3.5: An representation of a composite and equal size alternatives RVE (windows). apparent properties were based on two classic types of boundary conditions. Uniform displacement (also called kinematic (KUBC), essential, or Dirichlet), results in an upper bound. Uniform traction (also called static (SUBC), natural, or Neumann), results in

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an lower bound for the effective Young’s modulus [57]). If the volume element is big enough, the estimated upper and lower bounds coincide, and the apparent properties are identical to the effective ones [55,58].

Windowing methods are based on a surface integral version of the Hill condition Equa-tion 3.13, that can be rewritten as

Z Γ t(x) − hσin(x) T u(x) − hεix dΓ = 0, (3.14)

where t, u, n and x are the traction, displacement, normal and position vectors respect-ively. There are four ways to satisfy these conditions, three of them based on uniform boundary conditions [48, 59]. Uniform displacement, can be achieved by prescribing a constant strain ε0 _{= hεi on all boundaries of the volume element. This way, the right}

term on the Equation 3.14 is placed to zero,

u(x) = ε0x ∀x ∈Γ, (3.15)

where the superscript 0 denote a constant tensor. Uniform traction boundary con-ditions, can be applied by prescribing a given macroscopically homogeneous constant stress tensor σa_{= hσi on all faces of the volume element,}

t(x) = σ0n ∀x ∈Γ, (3.16)

it sets to zero the first term of the Equation 3.14 Hazanov and Huet, realized that is very difficult to reproduce this two types of load experimentally, especially the first one, and proposed a third type of uniform boundary conditions, Mixed Uniform Boundary Conditions (MUBC) [47,48]

[t(x) − σ0_n_]T_{[u(x) − ε}0_x_{] ∀x ∈ Γ,} _(3.17)

MUBC must be orthogonal and can be realized only in materials having at least ortho-tropic elastic symmetry. The fourth way is a special extension of the mixed boundary conditions proposed by Pahr and Zysset, named Periodicity Compatible Mixed Bound-ary Conditions (PMUBC) [48,55].

In the subject of the current work, windowing approaches have been used in a con-siderable number of studies. Non-periodic cellular arrangements, usually given by tomo-graphy or Voronoi diagrams, are subjected to these approaches. Silva et al. [57], used a finite element method to model a 2D random Voronoi cells, with uniform wall thickness, and found that the variability in the arrangement of the cell walls introduce a small change in the elastic constants of isotropic Voronoi structure. They also found that the structure-property relations for isotropic and anisotropic are on average no different from those for periodic honeycombs. Based on previous works, Chen et al. [60] used a 2D periodic Voronoi structure and, by introducing imperfections into the microstructure of the open-cell foams, they predicted an elliptical yield surface for elastoplastic foams un-der multi-axial loading. They investigated three different boundary conditions, KUBC, SUBC and periodic boundary conditions and concluded that, in order to produce faith-ful results, the periodic boundary must be employed [61]. Zhu et al. [62, 63] generated periodic, three-dimensional (3D), random samples with different degrees of irregularity to study how these cell irregularities affect the elastic properties of open-cells. They

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3.Modelling Cellular Materials 21 concluded that open-cell foams that are more irregular at constant relative density, have higher effective Young’s modulus and shear modulus, the bulk modulus is lower and the Poisson’s ratio is independent of the degree of irregularity [64]. Based on this work, Zhu and Windle [62] used the same 3D Voronoi method to investigate the influence of cell irregularity on the response to high strain compressions and found that, for low strains, strut bending and twisting dominate the mechanical response for irregular low density foams. However, for large compressive strains, strut buckling becomes the main mechan-ism of deformation. Gan et al. [61] used the methodology developed by Chen, to create three-dimensional Voronoi models to study the mechanical behaviour of linear elastic open-cell foams. Through finite element analysis, they evaluated the dependence of the Young’s modulus, Poisson’s ratio and bulk modulus of the foams on the relative density. They found that, for small relative density regime, the elastic constants predicted by Voronoi foam do not differ much when compared to Kelvin foam models. However, for high relative density, Kelvin models overestimate both Young’s and bulk modulus of random foams. More recently, Pahr and Zysset [55] studied the influence of boundary conditions on apparent elastic properties of cancellous bone. They extended the uni-form boundary conditions from Hazanov and Huet [59] to porous materials, proposed a new type of boundary conditions PMUBC and found them to be the superior choice in the case of nearly orthotropic bone. Even though windowing approaches are usually used on highly complex phase-arrangements, these non-periodic arrangements can also be modified to become periodic, being able to be subjected to unit-cell approaches.

3.3.2 Periodic Microfield Approaches

In the periodic microfield approaches, the inhomogeneous material is approximated by an infinite model with periodic phase arrangement. The local fields can be evaluated based on the behaviour of the representative unit-cells ranging from simple periodic inhomo-geneities to highly complex phase arrangements, by numerical or analytical methods. These unit-cell models can provide highly detailed information on the behaviour of the unit-cell local stress and strain fields.

Many studies regarding the application of these methods on cellular solids unit-cells have been revealed so far, and had also proven the utility of these methods in this matter. One of the earliest works to use finite element models on an equilateral tetrakaidecahedral structure was done by Simon and Gibson [65]. They investigated the effects of the material distribution between the edges and cell walls of 2D honeycombs and 3D kelvin foams, they also studied the influence of face curvature and corrugations on the stiffness and strength of the structures. They found that the distribution of material in the cell walls has little effect upon the Young’s modulus, and has a modest influence upon the uniaxial yield strength. They claim that at the joints of the honeycomb the bending moments has the highest values and so the presence of plateau borders has small influence on the young’s modulus.

The influence of wavy distortions in cell walls on stiffness of the structure was also studied by Grenestedt [66, 67]. He showed that such imperfections, have a large influ-ence upon the elastic properties. The influinflu-ence of different cell wall thickness has been investigated by Grenestedt and Bassinet [68], they conclude that has no affect on the elastic properties. They used two flat face Kelvin foam, Body Centred Cubic (BCC) and Face Centred Cubic (FCC) as a realistic approximation to a real closed foam, and found

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that the Young’s modulus predicted by these models was proportional to their density and the predicted Poisson’s ratio was on average 0.32.

Roberts and Garboczi [37] used finite element method to estimate the Young’s modu-lus of realistic random models of isotropic cellular solids, by applying periodic boundary conditions to the structure. They noticed that the results for Voronoi tessellations were in agreement with flat Kelvin models, also they studied the effect of deleting faces and found that if more than 70% of the cell faces are removed the dominant mechanism of deformation is edge bending. Sihn and Roy [69] modelled a tree dimensional carbon open foam to predict the effective properties of a carbon foam. They notice that the transverse and shear properties of the carbon foam should be improve rather than the longitudinal properties. They also found that effective Young’s modulus improves sig-nificantly by increasing the amount of material in the middle of the ligaments. Gong et al.[70] studied the open-cells response to uniaxial compression, experimentally, using 5 samples of uniform polyester foams, and numerically, modelling a Kelvin unit-cell. They concluded that Kelvin cell model developed replicates with some precision the initial elastic behaviour of the analysed foams. Daxner et al. [15] studied the usage of space-filling polyhedra, such as Kelvin cells and Weaire-Phelan partition, as models of solidified dry foams. The degree of anisotropy and effective properties were investigated. They found that for the same density range, Kelvin structures yield a larger degree of anisotropy, and consequently a larger direction-dependence of the effective elastic modu-lus. The average effective elastic modulus and the shape of yield surface in the 3D space of normal stress components is approximately the same for both structures. The same authors [35], generated a micromechanical finite element model, based on the Weaire-Phelan structure, for metallic open foams with hallow struts. The elastic properties were predicted by means of a unit-cell model and compared to experimental results. Due to the presence of imperfections, the experimental data for the effective stiffness overlapped the minimum values predicted. Buffel et al. [31] also used minimal surface energy ap-proach or the maximum space-filling polyhedra, Kelvin and Weaire-Phelan structures, to determine the elastic response of open-cells and to compare it with other methods and experimental data. And came to the same conclusions as Daxner et al. [15] but for open foams, the stiffness and Poisson’s ratio predictions as a function of density is approximately the same for the Kelvin and Weaire-Phelan structures.

Boundary Conditions

The drawbacks of the windowing methods are that the homogeneous displacement boundary condition overestimates the effective moduli while the homogeneous traction boundary condition underestimates the effective moduli. Also the application of homo-geneous displacement boundary conditions generally are not guaranteed to produce a periodic boundary traction. Similarly, periodic displacement at the boundaries is not guaranteed by using uniform traction boundary conditions [71]. Considering a unit-cell that is located at a large distance from the boundary of the inhomogeneous body, at the microscale the stress and strain fields conform based on the periodicity of the geometry. The stress and strain fields depend on the two variables macro and micro as exposed in the Equation 3.2. Depending on the macrovariable they can vary from one place to the other. However, their local variations are considered to be periodic, which means that this value has the same magnitude on the displacement fields u of all boundaries. [72]. In

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3.Modelling Cellular Materials 23 the periodic homogenization, the stress and strain fields, are decomposed into constant macroscopic contributions hεi and hσi, ε0_{(x) and σ}0_{(x) periodically varying microscopic}

fluctuations, as in the Equation 3.2. Here x denote the position vector of a point in a unit-cell. As stated by Suquet (1987) [71], these periodicity conditions on the boundary ∂Ω is

u(x) = hεix + u0(x), u0(x) periodic. (3.18) In the Equation 3.18, hεi are the average strains, u0 _{is the periodic part of the}

dis-placement components on the boundary surfaces. Depending on the load applied u0 _is

generally unknown. Notice that, since ε derives from u, the periodicity of u0_{(x), implies}

that the average of ε0_{(x) = 0. From the general condition 3.18, it is possible to obtain a}

more suitable form of periodic boundary conditions. For a cubic volume, it is understood that the displacements take identical values for a pair of points on the opposite boundary surfaces of the unit-cell. Considering x a position vector placed at the boundary of the unit-cell, and L the length of the unit-cell,

u(x + L) = hεi(x + L) + u0, (3.19)

u(x) = hεix + u0, (3.20)

as u0 _{is periodic, the difference between Equations 3.19 and 3.20 is}

∆(u) = u(x + L) − u(x). (3.21)

In the case of periodic boundary conditions, the boundary condition does not constrain individual points on the boundary, as it happens for Dirichlet boundary conditions, but relates the values between two or more different points. Considering ci "shift vector" a

linear combination of the periodicity vectors, the following periodic boundary condition is obtained [48, 71,72],

u(x + L) − u(x) = ci. (3.22)

As it happens for Dirichlet boundary conditions if ci = 0 the condition is called

ho-mogeneous, otherwise, if c 6= 0, the condition is non-hoho-mogeneous, which means that it depends on the imposition of a given state of deformation. In addition to the geometrical periodicity of the RUC, it is also necessary to guarantee that the deformed RUC is also periodic. Thus, to ensure cell-to-cell continuity, the opposite geometrical boundaries of a given cell have to be identical, both for the original and deformed states. The periodicity of the deformed RUC depends on the periodic boundary conditions applied.

In the present work the application of periodic boundary conditions for a paral-lelepiped in y1 ∈[0, y10], y2∈[0, y20], y3 ∈[0, y03] (Figure 3.6), can be defined as:

χ(0, y2, y3) = χ(y10, y2, y3),

χ(y1,0, y3) = χ(y1, y02, y3) and

χ(y1, y2,0) = χ(y1, y2, y03),

(3.23) Where χ is the characteristic displacement fields tensor. Displacements and rotations of an arbitrary point of the unit-cell must be fixed in order to prevent rigid body motion.

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By applying the above relations, all the vertices are forced to act equally, avoiding rigid body motion (Figure 3.6).

Figure 3.6: Representation of an original RUC and the respective periodic deformation.

When the unit-cell faces are coincident with the symmetry planes of the phase ar-rangement of the geometry, periodic boundary conditions can simplify to symmetry boundary conditions. This means that the degrees of freedom for three faces of the unit-cell, in case of a three dimensional RUC, are constrained and the displacement in this faces are only tangential, therefore enforcing the condition that a pair of faces must stay parallel throughout the deformation history (Figure 3.7) [48]. Symmetry boundary conditions are easy to apply, however can only handle uni-axial load cases.

Figure 3.7: Representation of the application of symmetry boundary conditions to a unit-cell.

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3.Modelling Cellular Materials 25 Asymptotic Expansion Homogenization

In this work the periodic boundary conditions were used in the Asymptotic Expansion Homogenization (AEH). While the homogenization by applying symmetry boundary conditions is fairly straightforward, the (AEH) is more complex to use. Consider a inhomogeneous material Ω with a periodic microstructure composed by representative unit-cells with an associated body Y. Assuming the existence of two separated dimen-sional scales, x and y, linked to material behaviour at the macroscale Ω and microscale Y levels, respectively. Being the relation between the length scales, the variables related to these fields functionally depend on x and y, where

y= x

. (3.24)

As a result of the relation, the functional dependence in y is periodic in the domain Y. This characteristic is often designated by Y-periodicity. The inhomogeneous micro-structural Y-periodicity is reflected on the fact that the elasticity tensor D is Y-periodic in y. However the material homogeneity at the macroscale results on the direct non-dependence of the tensor D relatively to the macroscale coordinate system, x. In this context the elasticity tensor component is

D= D(y). (3.25)

When bridging to the macroscale coordinate system, x, the microstructural inhomogen-eity manifests in a factor of −_{1 times lower than the characteristics of the domain Y.}

According with to the Equation 3.24, at the macroscale the tensor D can be denoted as D(x) = D

_x

. (3.26)

The superscript stands for the fact that the elasticity tensor D is Y-periodic on the macroscale coordinate system, x, indirectly depending on x. The microscale problem is solved on two main steps. The first is associated to the calculation of the characteristic displacement field tensor χ. The elementary strain and stress matrices are given by ε = Bu and σ = DBu, respectively where B is the element strain matrix, u is the vector of nodal displacements, and D is the matrix of material properties. Therefore the calculation of the corrector matrix χ [73,74]

Z

YeB

T_DB_{dY χ =}Z

YeB

T_D_{dY = F}D_, _(3.27)

where the script e corresponds to element quantities associated with the discretized FE domain of the unit-cell, namely the body Ye_{. The corrector is a matrix on contrary to}

the case of displacements in conventional elasticity. The second term of the Equation 3.27 is made of the columns of the load matrix FD_{. These columns are six load vectors,}

leading to the same number of system of equations that need to be solved [73, 75]. Effective Properties

In the second step of the microscale problem solving, the matrix χ is used to correct the homogenized elasticity properties, accounting for the microscale material distribution

Methods for the prediction of effective properties of metal foams

José Miguel

Redondo de Aquino

Methods for the prediction of effective properties

of metal foams

Metodologias de previsão de propriedades efetivas de

espu-mas metálicas

José Miguel

Redondo de Aquino

Methods for the prediction of effective properties

of metal foams

Metodologias de previsão de propriedades efetivas de

espu-mas metálicas

Contents

List of Tables

List of Figures

Chapter 1

Introduction

1.1

Objectives

1.2

Document Outline

Chapter 2

Foam Properties

2.1

Cellular Solids

2.2

Foam Structure

2.3

Relative Density

2.4

Mechanical Properties/Compression Behaviour

Chapter 3

Modelling Cellular Materials

3.1

Foam Models

3.2

Micromechanics of Materials

3.3

Micromechanical Methods for Material

Characteriza-tion